Descriptive Statistics

Modified True or False. If the statement is true, write T and if the statement if false , write F and explain why!


Q1. The values computed for a sample, arithmetic mean =-4.5, and root mean square= -4 are valid since the root mean square is always greater than or equal to the arithmetic mean.

Q2. Variances computed from a sample were sample variance = 0.4629, and population variance = 0.5141 are valid.

Q3. The variances computed with a sample size n = 10, population variance = 0.4629, and sample variance = 0.60, are valid.

Q4. Various means computed for a sample: harmonic mean, HM = 0.2, geometric mean, GM = 0.2, arithmetic mean, AM = .3,
root mean square, RMS = .4 are valid since they satisfy \(HM\le GM\le AM\le RMS \).

Q5. If the variance of a sample X is 5.3 then the variance of \(Y = -2 (X + 5)\) is \(2 \cdot 5.3 = 10.6\).

Answers:

1. F. The reasoning may sound valid but the root mean square is never negative!

2. F. Sample variance is always greater than population variance!

3. F. Sample varince may be obtained using the formula: $$\sigma_{n-1}^2= \frac{n}{n-1} \sigma_n^2$$. But $$\frac{10}{9} 0.4629 = 0.5141$$ which is not equal to 0.60.

4. Although the given values does satisfy the inequality for various means, whenever any two of the means are equal, then all means should be equal!

5. If X has the variance \(V_X\), then \(Y = a(X + b)\) will have variance $$a^2 V_X$$. Therefore, the varice of Y will be 4 times the variance of X or 4 (5.3)= 21.2.